For the negative results, consider a \(4\times 4\)  grid graph \(G\)  with
vertices \((x,y)\)  for \(1 \le x,y \le 4\) . Triangulate \(G\)  by edges
parallel to the diagonal. Then replace each edge by a “fat edge”.
For fan-crossing (fan-planar) graphs, a fat edge \(u v\)  is
\(K_7\)  with vertices \(u\)  and \(v\)  in the outer face of an embedding.
Call the resulting graph \(H\) . Binucci et al. [1]}
have shown that \(K_7\)  has an almost unique fan-planar embedding
with two vertices in the outer face, which results from fragments of
crossed edges, see Fig. REF . The result holds
for fan-crossing graphs, since configuration II [2]}
is not used, and for 2-planar graphs. Fat edges cannot be crossed
without violation. In case of 1-planar graphs, a fat edge is \(K_6\) ,
see Fig. REF , and an edge for planar graphs. In
consequence, any fan-crossing or 1- or 2-planar embedding of \(H\)  is
a grid with fat edges. For every fan-crossing (\(j\) -planar with
\(j\le 2\) ) drawing of \(H\) , there is a fat edge between the vertices
\(u\)  and \(v\)  at positions \((2,2)\)  and \((3,3)\) , which cannot be
crossed without violation. Hence, a path \(P\)  cannot be added between
\(u\)  and \(v\)  so that \(H \oplus P\)  is fan-crossing (\(j\) -planar with \(j\le 2\) ), if \(G\)  is fan-crossing (\(j\) -planar).
